The majority of existing literatures only investigates the total power consumption of the active RIS [1]}, [2]}.
However, the amplification circuit electronics have to work in their linear region where the output power increases linearly with the input power. In this work, considering the limited power magnification capability
due to the low cost of the amplifier, we also consider the element-wise power constraint, which reads as
\(a_{m}\left(\left|x_m\right|^{2}+ \sigma _v^2\right)\le p_{\mathrm {max},m}, ~~\forall m,\) 
